bayesian statistical approach to dark matter direct detection

Chiara Arina

Bols

hoi s

imul

atio

n, N

ASA

Bayesian statistical approach to dark matter direct detection experiments

PASCOS 201319th International Symposium on Particles, Strings and Cosmology

Taipei, November 20-26, 2013

1

Outline

• Bayesian analysis of direct detection data motivated by(i) Tension between experiments (4 hints of detection and exclusion bounds)(ii) Experimental systematics (e.g. Leff, quenching factors) and backgrounds(iii) Astrophysical uncertainties in both the halo parameters and velocity distribution

• Bayesian Evidence for model comparison and compatibility• Best scenario that accommodates XENON100 and the hints of detection (DAMA, CoGeNT, CDMS-Si, CRESST)

• Best particle physics scenario for hints of detection

• Quantitative measure of incompatibility between XENON100 and hints of detection

• Conclusions

C. Arina (IAP, Paris & GRAPPA Institute, UvA) - PASCOS 20132

• CA, J.Hamann and Y.Wong, JCAP 1109 (2011)• CA, J.Hamann, R.Trotta and Y.Wong, JCAP 1203 (2012)• CA, Phys.Rev.D86 (2012)• CA, arXiv: 1310.5718, invited review for special issue of PDU

Bayesian Inference framework

Likelihood(proper of each EXP)

PriorPosterior probabilityfunction (PDF)

data

�i

�k

theoretical model parametersnuisance parameters = astrophysics and systematics

� = {�1, ..., �n,�a, ...,�z}





data

�i

�k


� = {�1, ..., �n,�a, ...,�z}

Common prior choices that do not favour any parameter region





data

�i

�k


� = {�1, ..., �n,�a, ...,�z}

Posterior sampled with nested sampling techniques (MultiNest) given the likelihood and the priorand marginalized over nuisance parameters






data

�i

�k


� = {�1, ..., �n,�a, ...,�z}

Posterior sampled with nested sampling techniques (MultiNest) given the likelihood and the priorand marginalized over nuisance parameters



Profile Likelihood is prior independent (comparison with frequentist approach)

4 C. Arina (IAP, Paris & GRAPPA Institute, UvA) - PASCOS 2013

Bayesian Inference frameworkMarginalization over all nuisance/new physics parameters

Background and systematics

Astrophysical parameters(common to all exp)

Beyond elastic SI scattering(common to all exp)

Inference for constraining data , example with DAMA1D marginalized posterior PDFquenching factors (nuisance)

Matches with profile likelihood analysisC. Arina (IAP, Paris & GRAPPA Institute, UvA) - PASCOS 20135

log10(mDM) [GeV]

log 10

(σnSI

) [cm

2 ]

Posterior PDFElastic scattering

0 0.5 1 1.5 2 2.5 3−41.5

−41

−40.5

−40

−39.5

−39

2D marginal credible regions at 90 and 99% (shaded f(v)=NFW, solid blue f(v)= MB)

Inference for non constraining data: CDMS-Si


log10(mDM) [GeV]

log 10

(σnSI

) [cm

2 ]

Posterior pdf

1 1.5 2 2.5 3−44

−43

−42

−41

−40

−39

−38

log10(mDM) [GeV]

log 10

(σnSI

) [cm

2 ]

Profile likelihood

1 1.5 2 2.5 3−44

−43

−42

−41

−40

−39

−38

2D marginal credible regions at 68 and 90%for fixed astrophysics

• Likelihood follows a Poisson distribution with spectral information • 3 events seen with estimated bckg of 0.7: not constraining data

data from CDMS-Si collaboration arXiv:1304.4279

Inference for exclusion bounds: XENON100


Data are not constraining therefore the upper bound depends on the prior choice:

log10(mDM) [GeV]

log 10

(σnSI

) [cm

2 ]

Posterior pdfElastic Scattering

0.5 1 1.5 2 2.5 3−45

−44

−43

−42

−41

−40

−39

−38

Invariant exclusion bound based on the S signal with bayesian interpretation:

2D marginal credible regions at 90% +

from arXiv:1104.2549

- 2 events seen, likelihood follows a Poisson distribution - expected background of 1. +- 0.8, analytical marginalization - considered Poisson fluctuation below threshold

data from XENON100 collaboration, arXiv:1207.5988

Inference for elastic SI scattering


log10(mDM) [GeV]

log 10

(σnSI

) [cm

2 ]

Posterior PDF Elastic scattering

Marginalized astro (NFW)

1 1.5 2 2.5 3−45

−44

−43

−42

−41

−40

−39

−38

- The marginalization over astrophysics does not improve the compatibility between XENON100 and all detection hints- The XENON100 bound is less stringent at masses larger than 30 GeV than the one of the collaboration because of

the approximate likelihood - Same analysis can be done with LUX, more difficult to reconcile low mass regions, as its threshold is at 2 PE

arXiv:1207.5988profile likelihood analysisfixed astrophysics

Inference for isospin violating scattering


log10(mDM) [GeV]

log 10

(σnSI

) [cm

2 ]

Posterior PDFIsospin violating scatteringMarginalized astro (NFW)

1 1.5 2 2.5 3−44

−43

−42

−41

−40

−39

−38

−37

−36

- The extra parameter is not supported/constrained by current data

- The marginalization over the parameter causes a volume effect: detection regions becomes larger and the exclusion bound moves to the right

- Within the Bayesian approach the hint regions become compatible with the 90% CL of XENON100

- Inelastic and exothermic dark matter have same volume effect, however the agreement between detection regions and exclusion bounds is worst than isospin violating scenario

Assumption that interaction of WIMP with proton and neutron is of different strength:

Bayesian evidence for model comparison


Bayesian evidence1. model averaged likelihood2. contains notion of Occam’s razor principle3. used for model comparison and statistical test

Posterior pdf for a model:

�(M0) = �(M1)(non committal prior)

Bayes factor: ratio of model’s evidences

Empirical Jeffreys’ scale

Combined fit and model comparison


Combined fit pushes the quenching factor and the local standard at rest at corner values (1D pdf not flat anymore but peaked at qNa=0.6)

log10(mDM) [GeV]

log 10

(σnSI

) [cm

2 ]

Posterior PDF Elastic scattering


0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4−41

−40.5

−40

−39.5

−39

−38.5

−38

• Evidence between elastic and isospin violating scenarios is inconclusive• Inelastic and exothermic moderately disfavored with respect to elastic SI

(In)Compatibility between XENON100 and detection hints ?


Nevents

ln L

(Nev

ents |

D)

1 σ

2 σ

3 σ

Observed

0 10 20 30 40 50 60 70 80 90 100−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

0Consider a data set which is given by two subsets:1. = this is the data set which is taken as true, hence fixed2. = data set that we want to test, that is we want to quantify if it is compatible with

Elastic scattering

Isospin violating framework: likelihood ratio in data space gives incompatibility at

Summary

• Bayesian approach for XENON100, DAMA, CoGeNT, CRESST and CDMSI-Si data with marginalization over the systematics and nuisance parameters characteristic of each experiment (can be applied to LUX, similar to XENON100 procedure)

• Inclusion of velocity distributions arising from DM density profile and marginalization over astrophysical variables (NFW)

• Difficult to reconcile at 90% CL all detection hints and XENON100

• Going beyond the elastic SI scattering (isospin violating, inelastic and exothermic scattering) ameliorates the compatibility between experiments: the additional physics parameter is not constrained by the current data

• Astrophysical uncertainties can not be yet constrained by direct detection experiment alone (however combined fit can constrain astrophysics)

• Combined fit implies large value of the quenching factor on Sodium for DAMA and small local standard of rest velocity

• For hints of detection the elastic and isospin violating scenarios have the strongest support form the data; isospin violating framework ameliorate the compatibility between hints of detection and exclusion bounds


Back up slides


Inference for inelastic/exothermic SI scattering


log10(mDM) [GeV]

log 10

(σnSI

) [cm

2 ]

Posterior PDFExothermic scattering


0.5 1 1.5 2 2.5 3−44

−43.5

−43

−42.5

−42

−41.5

−41

−40.5

−40

−39.5

−39

log10(mDM) [GeV]

log 10

(σSI n

) [cm

2 ]

Posterior PDFInelastic scatteringMarginalized astro (NFW)

1 1.5 2 2.5 3−44

−43

−42

−41

−40

−39

−38

−37

−36

16

Changing the WIMP physics interaction Isospin violating interaction

• Assumption that interaction of WIMP with proton and neutron is of different strength:

• Defined a mean SI cross-section with an effective couplings to nuclei:

Feng et al. ’11

(Schwetz, Zupan ’11)

• Example of realization in WIMPs model: The couplings neutralino-squark-quark violate isospin, however in the most common scenarios they are not the dominant contributions to elastic scattering

• Other possibilities: long range interactions, inelastic scattering, spin-dependent interaction


17

Inelastic scenario

q q

DM DM*DMq ! DM? q

• If the splitting is positive the DM scatters into an heavier state: kinematic condition implies that the scatter occurs most probably with heavy nuclei (hence more sensitive to heavy WIMPs)

• If the splitting is negative exothermic Dark Matter, it decays into a lighter states and light target are favoured

Tucker-Smith, Weiner ’01

�m ⌘ (mDM? �mDM) = �

� ⇠ O(100 keV)


18

SI elastic scattering scenario CDSM-Si

lnLCDMSSi =

"3X

i=1

P1(S +B)

#+

2

4X

j

P0(S +B)

3

5+ lnLbck


1D marginalized posterior PDF for all parameters:

data from CDMS-Si collaboration arXiv:1304.4279

19

Combined fit more details


qNa0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1combined fit pushes the quenching factor and the local standard at rest at corner values (1D pdf not flat anymore but peaked at qNa=0.6)

Combined (elastic SI)

Single experiments

20

Combined fit more details


δ (keV)

Post

erio

r PD

F

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

fn/fp

Post

erio

r PD

F

−2 −1.5 −1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

log10(mDM) (GeV)

log 10

(σnSI

) (cm

2 )

NFW density profile

Combined set of data

0 0.5 1 1.5−42

−41.5

−41

−40.5

−40

−39.5

−39

−38.5

−38

−37.5

−37

21

DAMA and CoGeNT, combined fit: hidden directions behavior

- the larger qNa the smaller the WIMP mass- low mass region is independent on qI

- similar behavior for the DM density at the sun position- less sensitive to the escape velocity value


XENON100

- S = 2 (seen events), likelihood follows a Poisson distribution - B = 1. +- 0.8- Total exposure 2323.7 kg days

Aprile et al. arXiv:1104.2549XENON100 collaboration, arXiv:1207.5988

- Scintillation efficiency is a systematic of the experimental set-up- treated as nuisance parameter with truncated gaussian prior and marginalized over


XENON100

23

conversion between keVnr and PE

All the likelihoods are normalized such that if the background matches exactly the number of observed events


24

CoGeNT 2011Germanium cryogenic detectordetector mass 0.33 kglive time 442 daystotal exposure 145.86 kg days

- Data analysis and binning follow arXiv:1106.0650 [astro-ph.CO]- Radioactive peaks subtracted as prescribed by the collaboration- Analysis of the total rate with a background (27 bins)- Analysis of the modulated rate without background in 3 energy bins- All data are corrected by the efficiency factor, ranging from 0.7 to 0.82

Total rate : 27 bins of width 0.1 keVee energy range 0.5- 3.2 keVee

Modulated rate:

3 nuisance parameters for the non modulating background

quenching factor:


CoGeNT

DE1 = H0.5-0.9L keVee

0 100 200 300 400 50020

30

40

50

60

t HdaysL

Countsê3

0days


0 100 200 300 400 50020

40

60

80

100

t HdaysL

Countsê3

0days


0 100 200 300 400 50010

20

30

40

50

60

t HdaysL

Countsê3

0days

2.3�Modulation: from to 1.6�

Aalseth et al. arXiv:1106.0650

CA, J.Hamann, R.Trotta & Y.Wong arXiv:1111.3238 [hep-ph];

Ge detector, 146 kg days Very low threshold:0.4 keVee = 2.7 keV

25

Gaussian likelihood

• Background1. does not modulate, included only for the total rate2. constant + exponential background (mimic surface events)3. 3 nuisance parameters

• Radioactive peaks subtracted


CoGeNT 2011

26

Data analysisRadioactive peaks

arXiv:1106.0650


CRESST-II Angloher et al., arXiv:1109.0702evidence at 4 sigma

- 8 detector modules made by CaWO4 crystals (multi-target detectors)- scintillation + ionization to disentangle background

(e, n, alpha, decays of Pb isotopes)- exposure of 730 kg days- S = 67 events (background can account only for 65% of S)

light elements are more sensitive to light particles and viceversa!

27

Likelihood Poisson distribution1. Total counts in each module2. Global spectral information3. Background as nuisance parameters: 3 nuisance parameters


CRESST-II

28

Angloher et al., arXiv:1109.0702

- 8 detector module made by CaWO4 crystals (multi-target detectors)- scintillation + ionization to disentangle background (e, n, alpha, decays of Pb isotopes)- exposure of 730 kg days- S = 67 events (background can account only for 65% of N)


(In)Compatibility test

29

• We assume that:1. The fixed set is:2. The result to be tested is the number of events seen in the XENON100 detector

We find that elastic and isospin violating models are incompatible at 2 sigma level, while inelastic scattering is within 1 sigma

Inelastic

Isospin violating

Elastic with Nmax=60

Elastic wit Nmax=100


30

CoGeNT and combined fit disfavour inelastic scattering because the excess is in the low energy region and it prefers light WIMP masses

Which is the best model that accounts for the excessat low WIMP mass?

Elastic, inelastic and isospin violating models are nested models:Inelastic reduces to elastic for Isospin violating reduces to elastic for fn/fp = 1

� = 0


31

Parameter inference 3D: inelastic case

0 0.5 1 1.5 2 2.5 3−44

−43

−42

−41

−40

−39

−38

−37

−36

δ (keV)

log10(mDM) (GeV)

log 10

(σnSI

) (cm

2 )

0

2

4

6

8

10

12

14

16

18

20

0 0.5 1 1.5 2 2.5 3−44

−43

−42

−41

−40

−39

−38

−37

−36

δ (keV)

log10(mDM) (GeV)

log 10

(σn) (

cm2 )

0

20

40

60

80

100

120

140

160

180

200

0 0.5 1 1.5 2 2.5 3−44

−43

−42

−41

−40

−39

−38

−37

−36

δ (keV)

log10(mDM) (GeV)

log 10

(σn) (

cm2 )

0

20

40

60

80

100

120

140

160

180

200


32

Parameter inference 3D: isospin violating case

0 1 2 3−46

−44

−42

−40

−38

−36

fn/fp

log10(mDM) (GeV)

log 10

(σnSI

) (cm

2 )

−2

−1.5

−1

−0.5

0

0.5

1

0 1 2 3−44

−43

−42

−41

−40

−39

−38

−37

−36

fn/fp

log10(mDM) (GeV)

log 10

(σn) (

cm2 )

−2

−1.5

−1

−0.5

0

0.5

1

0 1 2 3−44

−43

−42

−41

−40

−39

−38

−37

−36

fn/fp

log10(mDM) (GeV)

log 10

(σn) (

cm2 )

−2

−1.5

−1

−0.5

0

0.5

1


33

Construction of DM velocity distribution (1)

DD depends on the distribution function (DF) at the sun position arising from the WIMPs phase-space distribution

• f(v) is a function of the gravitational potential (including baryon contribution)• f(v) is a function of the DM density profile

• DF obtained inverting the above equation • Symmetries assumed: density profile spherically symmetric and f(v) isotropic -> DF only function of the energy


34

Likelihood for astrophysical observables (nuisance parameters for ALL EXP)

The profiles mostly differ near the galactic center, at the sun position they give similar behavior for f(v)In what follow only shown comparison between NFW and SMH

Rü

0.001 0.01 0.1 1 10 1000.001

0.1

10

1000

r HkpcL

r DMHGe

Vêcm

3 LSpherically symmetric DM density profiles :

• NFW• Einasto• Cored Isothermal• Burkert

�DM = �DM(cvir,Mvir)

Construction of DM velocity distribution (2)


35

Sensitivity analysis

lnB2a = �1.06

• lnB of 1a:2a is now 3.11 instead of 5.21, still moderate evidence

• Results are robust from a Bayesian point of view!

For nested models with parameter priors separable the Savage Dickey density ratio (SDDR) gives an analytical estimate of the effect on lnB changing the width of the prior

-0.5 0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

2.0

2.5

Prior width

pHq»M1Lno

rmalized

EXAMPLE

marginal posterior pdfs, computed at fixed value of the parameters

marginal normalized prior densitycomputed at fixed value of

36

Velocity distribution from DM density profile

Assuming equilibrium between gravitational force and pressure:

Eddigton formula for spherically symmetric DM density profiles that lead to isotropic f(v)

Poisson equation for the gravitational potential including contribution from the bulge and disk:

The velocity distribution is translated to the reference frame of the Earth:


37

DM density profiles


Theoretical predictions for elastic spin-independent scattering off nucleus

Differential rate

Modulated rate


Annual ModulationDrukier, Freese and Spergel ’86, Freese, Frieman and Gould ’88

In the Earth’s rest frame the DM velocity distribution acquires a time dependence, which follows a sinusoidal behavior

Signature of WIMP recoil in the detector

v2 = |�v⇥ + �v�|2

v� = |�v⇥ + �v��,rot|

= v⇥ + v⇤⇤�,rot cos � cos[2⇥(t� t0)/T ]

� = 60�

effect of O(10%)

Bernabei et al. arXiv:1002.1028


40

CoGeNT modulation


41

Comparison between 5 phenomenological models that describe a sinusoidal modulation:

Is there evidence for DM modulation in CoGeNT?


0 100 200 300 400 50020

30

40

50

60

t HdaysL

Countsê3

0days


0 100 200 300 400 50020

40

60

80

100

t HdaysL

Countsê3

0days


0 100 200 300 400 50010

20

30

40

50

60

t HdaysLCountsê3

0days


42

-6 -4 -2 0 2 4 6ln(B)

0

1a

1b

2a

2bΔE1ΔE2ΔE3All ΔE

stron

g ev

iden

ce fo

r M0

mod

erat

e ev

iden

ce fo

r M0

wea

k ev

iden

ce fo

r M0

wea

k ev

iden

ce a

gain

st M

0

mod

erat

e ev

iden

ce a

gain

st M

0

stron

g ev

iden

ce a

gain

st M

0

-2.16

-1.88

0.88

0.57

1.49

-0.013

-0.98

0.61

-2.80

0.0048

-1.89

2.05

-3.16

1.50

-4.25

Non-DM T free

Non-DMannual

Consistent DM

Pheno-DM

Nomodulation

Bayes factor: results for model comparison


43

Classical p-valuesprobability of obtaining more extreme data than observed assuming the null hypothesis is correct and NOT probability for hypothesis

test statistics for nested models if1. additional dof distributed as a gaussian2. unbounded likelihood3. all additional dof identifiable under the null


43


test statistics for nested models if1. additional dof distributed as a gaussian2. unbounded likelihood3. all additional dof identifiable under the nullx


43


Chernoff’s theorem

� =N�

i=0

2��

��

i

�p(�2

i > ��2e�)



43



� =N�

i=0

2��

��

i

�p(�2

i > ��2e�)


2.3�


43



� =N�

i=0

2��

��

i

�p(�2

i > ��2e�)


1.6�

2.3�


43



� =N�

i=0

2��

��

i

�p(�2

i > ��2e�)

test statistics for nested models if1. additional dof distributed as a gaussian2. unbounded likelihood3. all additional dof identifiable under the nullxx

1.6�

2.3�


43



� =N�

i=0

2��

��

i

�p(�2

i > ��2e�)

test statistics for nested models if1. additional dof distributed as a gaussian2. unbounded likelihood3. all additional dof identifiable under the nullxx

Rely on Monte Carlo simulation formapping the t statistic into p-values

1.6�

2.3�


C. Arina (IAP, Paris & GRAPPA Institute, UvA) - PASCOS 2013C. Arina (IAP, Paris & GRAPPA Institute, UvA) - PASCOS 201344

Similar behavior for the All bin case: the inference is driven by bin 2

Parameter inference: amplitude of

45

Parameter inference: phase and period


46

Locally anisotropic DM velocity Ellipsoidal, triaxial DM halo model gives rise to a triaxial gaussian velocity distribution:


47

Background CoGeNTPriors on the fractional modulated amplitude predicted from configurations of DM mass and sigma that account for the CoGeNT total rate R(t) = S(t) + B

Background:1. does not modulate, included only for the total rate2. constant + exponential background (mimic surface events)

J. Collar talk @ TAUP 2011.


48

Model 1b: consistent DM Priors on the fractional modulated amplitude predicted from configurations of DM mass and sigma that account for the CoGeNT total rate R(t) = S(t) + B

Sm =R(tmax)�R(tmin)R(tmax) + R(tmin)


49

Annual modulation in CoGeNT and in CDMS collaboration, Z. Amhed et al., arXiv:1203.1309 [astro-ph.CO]

• 214 kg days exposure•requirement that event scatter only once in the detector• 5 keVnr as threshold• No evidence for annual modulation

Amplitudes larger than are excluded at


bayesian statistical approach to dark matter direct detection

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